 Kaj je inbečil, da je vzlušenje vzlušenje v zelo, na kateri, iko je prijevili problemov, vzlušenje, da je na vsev zelo vzlušenje, in zelo, ki je zelo povodil na taj del, razradi je noga. To je začala povod, iz Misha Grishka, ki je odradi v 2014. Tukaj je tukaj vse prekast, ker je tukaj tukaj atmosferične, konvektivne vse. Tukaj simulacija je skupnila v Spalm LS. Tukaj je za Pola region. Tukaj je tukaj tukaj konvektivne vse, kaj je zelo vse, kaj je zelo vse, tukaj je zelo vse, tukaj je zelo vse. Pozivim sem o ta delovnije vse navradi, vse je bljev, vse je vse blj, je vse je bilo komentan muž. Vse je bilo, da je vse vse, zelo vse je zelo vse, zelo vse je vse vse, vse vse je potenšiala temperatura, sasto, ki je poslednja z S-dajta z Rasa Reota, din S-dajta z Vagu, z Alfred Vagenirjučkih institutov, z Jorka Hartmann. Zelo sem zdenljela, da se večo različno obradi v figli, je z mimo količom, nekaj obradi in izvaj, Ok. Vse je... Geostrophic is forced by wind from above and you have in free atmosphere, geostrophic wind so called, which is caused by pressure horizontal gradient. Yeah. And due to carry-olis you have turning and V component appear. Ok. And here you also must have in mind that measurements, this is instantaneous fields, while the dns, oops, sorry, while dns, this is blue and Ls, here they are nearly coincide. They are horizontally average over domain of simulations. So next step. Now we see deviations from this mean, which was shown before. Here it's a potential temperature fluctuations. They here with prime. And this is site view. You have updrafts from below and downdrafts from group from above. It's data and here the view from above on this line, in the middle of boundary layer. And you can see that wind in this direction, so you can see the organization of plumes in some sort of strings. And temperature in vertical velocity you have, this is so called organized covariance structure or simply plumes. We just here they are small size in surface layer. After that they growing in size and in the middle they are represented like this. So now all standing questions. I must stress that I will talk only about one decay, if question arise about multidimensional features, I just will comment. And here there are, what does not work? There is no downgradian heat transport. There are no downgradian transport for third order moments, okay? And return to isotropy also does not work. As in homogenous turbulence. And finally famous milionček of hypothesis, sometimes called quasi normality assumption in DQNM framework and so on, also does not work. So, because all these feature in high order closure models, breakage means average, ensemble average, and in LS it's area averages. So, because in high order closure moment models, everything comes from milionček of hypothesis. It's the same for kinetic equation for closure models. And I will just analyze this only, this problem. Milionček of hypothesis was discovered in 41. And story of discovering of milionček of hypothesis, ah, it states that in all turbulence models, four order moments can be considered as gaussian, while third order moments completely non gaussian can be. And history of discovering of this is known because still survive people who was present in process of discovering that milionček of came to the lecture and Kalmagorov make his lecture, not lecture seminar. Small milionček of came, he was excited and tell that closure does not work, equations which he averaged non linear, now his talks does not work, and Kalmagorov turned back and tell making gaussian. And after that everything can start to work and so here I will demonstrate how milionček of hypothesis is presented in real signals. This is horizontal wind fluctuations. For them, of course gaussianality is working because it's some sort of nearly homogeneous and you can see. So what is shown here, here is the signal. Typical here it's pdf, gray one is gaussian. So you can see that deviation from gaussian is not very huge, actually very small and here skater plot which show gaussian behavior versus real measurements from this signal and everything is working perfect. But life is not so good for temperature fluctuations. Here you immediately can see huge deviation here from the gaussian behavior. And also this is clearly seen in pdf, you don't need even to plot milionček of curve by looking simply on pdf with gaussian. So we can expect them. So next step, this is what was known. In 2002 me and York Hartmann, we made some very simple trick. We just decided, okay, we will take, I talking about temperature, but the same was done for velocity components also. We take gaussian and we take very simple for delta model, which is called sometimes mass flux, sometimes cooking structure model and so on, but I will explain later what is this. And after that we suggest the simple, simple linear interpolation between two regimes that we tell everything is combined from very skewed gaussian turbulence, non gaussian turbulence. This is skewness, which is definition standard, nothing new here. And so we tested this trick and you see all these points move exactly on this line. So this is a new closure, which was called refined milionček of hypothesis. Why it's refined? Because here these two terms, this first term, it's exactly coincide with milionček of and skewness correction, it's refinement. Okay. So and we become happy with this because it works. And at the same time, we looked at some next step. If it works for refined milionček of hypothesis, we just made the trick that, okay, in turbulence it's allowed to make additional fitting constant and we just extended milionček of hypothesis by adding here ten fitting constant for five moments. And here there is the exercise. There are two mistakes in this formula. Here must be W of skewness and here also W of skewness because it's a higher order in velocities. So it's paste and copy and paste mistake. It's very good question because left hand side quadratic and here quadratic and skewness appear in square because it can appear without square in high order moments, I will show why. But here it's because due to symmetry simply. Please replace W to minus W, okay, in first equation. Right hand side. Okay. Yeah, yeah. Okay. So we was happy and decided that refined milionček of is better than extended milionček of. And now what happens in ten years after? Martin Losch made the simulation for deep ocean convection. Okay. And find out this functional form. Perfect. Linear interpolation work. After that, Friedrich Kupke made the simulations with compressible LS. The plumes have their speed near the Mach number. Okay. Very high. Kupke and Robinson. And they simulated sun and stars. After that, Lenchow made the measurements, ledar measurements in bold colorada. And Peter Sullivan made the LS simulations. It also worked. And recently it's not full set and I'm sorry if I not mentioned somebody. But finally, Vagoud made the DNS simulations, which I already shown. And this is also confirmed this functional form. And final step was made by Mironov in Močilske, who implemented this model in this parameterization in weather prediction cosmo model. And it also worked. So if such situation that then we have the functional form, okay, but all the studies reveal that actually turbulence moments can be as subgaussian, as supergaussian. That means not always free. It can be larger than free in zero skinous limit and can be smaller. So we decided to make something. So this is a question. What would be minimalistic model on gaussian, of course? And which can explain all this feature, which I already mentioned. And how to do this? What would be more or less universal description? And approaches is obvious. There are bulk models, there are runs models, first, second, third order. And ad hoc models. And also last one is LS and DNS models. So we just concentrate on ad hoc model. This is assumed distribution higher order closure model. It consists from several steps. They are obvious. Assume PDF, depending upon a parameter, solve for parameter in terms of irreducible PDF. Find higher order moments in terms of irreducible model moments. Test results against measurements LS DNS. If it does not work, repeat again and repeat again again before you reach some reasonable answer. So we quantify, we just decided to follow this problem. Here is a method from Marka Marigola. So it's a realistic signal. This corresponds to this. After that we come to Cubism of Pablo Picasso by simply making projections to PDF description. And finally we observe that we can approximate actual PDF by updraft motions. The velocity is only positive downdraft motions, then velocity negative. And some small scale background, they are around zero. And next step, most promising for analytical studies minimalism. Then you just move by replacing by delta functions for updraft, downdraft, and small scale turbulence. In analytical representation, everything looks very simple. You have PDF. Here I wrote for simplicity be variate case. There you have one, two, three, four for cognitive structure terms and one for background for small scale turbulence. W up, here it's updraft. They can be hot or cold. So you split, you have nine parameters. And for trivariate 15 and for quadrivariate 25 in all these cases we solved the problem. But I will talk only about variate for simplicity. So, what next? Next we have the equations for moments. They are looks like this. Any of these moments can be chosen as irreducible. But simplest choice, it's like this. Normalization conditions to zero for mean field, variances, flux, and two further moments, vertical velocity and temperature. They really basis for all future. So, fortunately it has nice solution, analytical solution for all moments. And structure, it's similar to Gaussian, but a little bit more complex. Here it's variances in corresponding power. Here, polynomial in correlation coefficients. These coefficients, which depends upon skevenesses. And ps, it's, you remember, it's 1 minus probability of background. Ops, sorry. And this depends upon ps, all these closure as a parameter. It's like fitting parameter. Area coverage has sense physical, has clear physical meaning. Area coverage in turbulence in nice coherent structures. And after that, we moved further. This is irreducible moments. This is measurements from aircraft campaigns. Here LAS, to which we feed our theory. And here is DNS, which feed data better. But we decided not to have many discussions with people about applicability of DNS. Here is complex situation to feed to LAS it easier to explain people because subgrid is more or less normal and everybody knows how it works. So, we did this heating. And now I go to result. This is now without mistakes formula. Here is our closure. This coefficient depends upon concentration of coherent structures. So, what we did actually, we matched the moments with physical morphology of coherent structure. If you know morphology, you can tell something about the closures. And now you see how it works. Here nearly perfect. The response to LAS data. Here parameter, there is no background at all. Here millončikov, refined millončikov hypothesis. P equal one third. And here then we increase or decrease PS concentration of coherent structure to 0.25. You see here. And physically everything is very nicely tractable if you have more coherent structure, more fluctuations. So, there are more large scale and you have more contributions so more moments must be larger. And rest of my talk is very simple. This is predicted moments for correlation between heat flux and variances. I have in mind W theta prime with W, for example here, W prime square. And everywhere have nice formulas depending on comparison, have very nice correspondence between data and our results. And next, we did the same for skewness variance calculation. And you see, now we come to answer to Simeon question. Here you see it's proportional, sorry, they are proportional to first power of skewnesses because they are not even but odd models. And after that, this is the end. This is summary. I finished. This is summary, technical summary. Technical summary consists of several statements. New exactly solvable model is suggested. High possesses, refined millions of hypothesis is proven and extended explained by fluctuations by this morphology of flow and the accuracy is very, very, very good and not bigger than skater which exist in numerical model due to subgrid scale closure for LS in comparison with DNF. Ok, here is romantic conclusion, philosophical. There always means that here you can see the perspective which is coming. And finally, thank you for your kind attention. And here you can read that there are two pessimistic and optimistic view of everything what was done. PCMist will tell, ok, better have no at all fitting constant and optimist tell, oh, we have reserve for 128. That's all. Spectrum, ok. There are two types of models. One model is called based spectrum models is called two point closure models, ok. But here it's one point closure models only. So spectrum is beyond the scope of such kind of models. They operate with much more simple feature, integral of spectrum, ok. Turbulence in recent understanding consists from three components. First component, mean field, second component, semi-organized structure and third homogenous isotropic turbulence of background. Here it's a mixture of semi-organized structure with small scale isotropic background. So we don't need actually spectra. Spectra not good representation for such situation. It fails completely. Yeah, yeah, yeah. Why it fails? Because you have some self-organization in the system. And spectrum representation. It suggests you quasi homogenous randomization of phases and something like this. But it's not necessary, it doesn't work. You must write something like simple closure equations plus additional equations for morphology of structure which are present there. In this case, concentration of coherent structure is a good parameter.